Trigonometric Graphs

From trigonometric ratios to functions

Sine, cosine and tangent can be treated as functions. The input is an angle $x$, measured here in degrees, and the output is a number. For example,

$\sin 30°=0.5,$

so $(30°,0.5)$ is a point on the graph of $y=\sin x$. The horizontal coordinate is an angle; the vertical coordinate is a ratio, so it has no unit.

The graphs extend the familiar right-angled-triangle ratios beyond acute angles. They describe negative angles, angles greater than 360° and angles of any size.

Why sine and cosine form waves

Imagine a point moving anticlockwise around a circle of radius 1. After a turn of $x$ degrees, its horizontal coordinate is $\cos x$ and its vertical coordinate is $\sin x$.

Neither coordinate can exceed 1 or be less than −1, so both functions have range

$-1\leq y\leq 1.$

After one complete turn of 360°, the point returns to its starting position. Therefore

$\sin(x+360°)=\sin x,\qquad \cos(x+360°)=\cos x.$

Sine and cosine both have period 360° and amplitude 1.

The sine graph

Over one complete cycle, $y=\sin x$ passes through

$(0°,0),\ (90°,1),\ (180°,0),\ (270°,-1),\ (360°,0).$

Starting at the origin, it rises smoothly to a maximum, falls through the horizontal axis, reaches a minimum and returns to the axis. The same pattern continues in both directions. The graph must be a smooth curve rather than straight segments joining the points.

Sine also has rotational symmetry about the origin:

$\sin(-x)=-\sin x.$

The cosine graph

Over one cycle, $y=\cos x$ passes through

$(0°,1),\ (90°,0),\ (180°,-1),\ (270°,0),\ (360°,1).$

Cosine has the same wave shape, amplitude and period as sine, but begins at its maximum when $x=0°$. It has reflection symmetry in the vertical axis:

$\cos(-x)=\cos x.$

The sine and cosine graphs intersect wherever their outputs are equal. Between 0° and 360°, this occurs at 45° and 225°.

The tangent graph

The graph of $y=\tan x$ consists of separate increasing branches rather than a bounded wave. Its central branch contains

$(-45°,-1),\ (0°,0),\ (45°,1).$

Tangent is undefined at 90° and at every further half-turn in either direction. The vertical asymptotes are

$x=90°+180°n,$

where $n$ is an integer. Each branch approaches its asymptotes, but the graph never touches or crosses them. Branches on opposite sides of an asymptote must not be joined.

Between consecutive asymptotes, a branch increases from increasingly large negative values to increasingly large positive values. Tangent has no maximum or minimum and its range is all real numbers. Its pattern repeats every 180°:

$\tan(x+180°)=\tan x.$

Its zeros occur at integer multiples of 180°.

Sketching the graphs

For sine or cosine:

1. Mark multiples of 90° on the horizontal axis.
2. Plot the five principal points in each 360° cycle.
3. Join them with a smooth wave.
4. Repeat the pattern when the interval extends beyond one cycle.

For tangent:

1. Draw dashed vertical asymptotes, starting at 90° and repeating every 180°.
2. Plot points such as $(0°,0)$ and $(45°,1)$.
3. Draw a separate, smooth increasing branch between each pair of asymptotes.
4. Leave every asymptote angle undefined.

A table of calculator values can provide extra points, but the calculator must be set to degrees.

Recognising and interpreting the graphs

An unlabelled graph can be identified from its defining features:

• sine is a bounded wave passing through the origin;
• cosine is a bounded wave with value 1 at 0°;
• tangent has increasing branches separated by vertical asymptotes.

A point on a graph represents a trigonometric statement. For example, $(210°,-0.5)$ on the sine graph means

$\sin 210°=-0.5.$

To solve an equation such as $\sin x=-0.5$ graphically, draw the horizontal line $y=-0.5$, locate every intersection with the sine curve and read the corresponding angles. Keep only solutions in the interval stated in the question.

A graph can also show that no solution exists. Because sine and cosine remain between −1 and 1, the equation $\cos x=1.4$ has no real solution.

Recognise the shapes

• $y=\sin x$: smooth wave through $(0°,0)$; maximum at $(90°,1)$ and minimum at .